1. Field of the Invention
This invention relates to a method and apparatus for analyzing the propagation of an optical field between planes that are tilted or offset with respect to each other, and to analysis of optical systems having elements in tilted or offset planes.
2. Background of the Invention
Today, in modeling, the effect of diffraction on the optical field becomes more and more important as the demand for more accurate models increases. Commercial software in general takes two different approaches in dealing with diffraction. Some commercial software takes a geometrical optics approach and takes diffraction into account only artificially by imposing the Frounhoffer diffraction pattern of the pupil on the plane of interest. Some other programs can propagate scalar fields using the Fresnel diffraction approximation. Between parallel planes such programs can use Fast Fourier Transform (FFT) techniques, but when it comes to propagation between tilted planes they resort to numerical integration. Numerical integration for two dimensional problems, depending on the window size, can be very time consuming as compared to a Fourier transform approach.
With the advance of integrated optical systems there is a need to propagate fields between tilted planes, since planes containing the optical elements are defined by the crystal structures in which these integrated optical systems are made. As is the case in silicon based optical integrated circuit technology, most crystal structures can be etched efficiently only in certain crystal directions, and these direction are not necessarily orthogonal to the optical axis of the system. Another reason to have tilted planes in an optical system is to distribute the refractive power of the plane between the optical element and the tilted surface.
Thus it is clear that there is a need to propagate optical fields between tilted planes using scalar diffraction theory both rapidly and accurately. The Rayleigh-Sommerfeld diffraction integral is as accurate as one can get for a scalar treatment. The problem with the Rayleigh-Sommerfeld integral is that when the input and output planes are tilted with respect to each other, one has had to resort to numerical integration since there has not been a way known to use Fourier transform techniques to speed up the calculation.
The problem of tilted planes has been previously treated. Ganci [S. Ganci, Eur. J. Phys. 2:158 (1981)] looked at the specific problem in which a plane wave, tilted with respect to a plane containing a slit, was diffracted onto another plane parallel to the slit plane under the Frounhoffer approximation. Later Patorski [K. Patorski, Optica Acta 30, 673(1983)] further calculated the Frounhoffer intensity pattern of the tilted slit plane on a plane that was perpendicular to the initial plane wave propagation direction. Rabal, Bolognini and Sicre's paper [H. J. Rabal, N. Bolognini, E. E. Sicre, Optica Acta 32, 1309 (1985)] generalized the previously mentioned work and concluded that the intensity pattern due to Frounhoffer diffraction from a tilted plane onto another plane perpendicular to initial beam propagation direction can be calculated by taking the Fourier transform of the tilted plane transmission function in its own coordinate system.
The first paper that addressed the question of finding the diffraction pattern of a tilted plane under the Fresnel approximation came from Leseberg and Frere [D. Leseberg and C. Frere, Applied Optics 27, 3020 (1988)]. Essentially, Leseberg and Frere showed that approach taken by Rabal et al. can be generalized by taking the Fourier transform of the tilted plane phase function times the quadratic phase factor only at spatial frequency components that have a special relation to the spatial coordinates in the output plane, as expressed by Rabal et al. Later Frere and Leseberg suggested another formulation [C. Frere and D. Leseberg, Applied Optics 28, 2422 (1989)] to approximate diffraction patterns of off-axis tilted objects. Finally Tommasi and Bianco proposed a technique [T. Tommasi and B. Bianco, Optics Letters 17, 556 (1992)] to find the relation between the plane wave spectrum of the same field with respect to two coordinate systems only rotated with respect each other, and they used this approach to calculate the computer generated holograms of off-axis objects [T. Tommasi and B. Bianco, J. Opt. Soc. Am. A. 10, 299 (1993)]. Bianco and Tommasi also applied their approach to model space variant optical interconnects [B. Bianco, T. Tommasi, Applied Optics 34, 7573 (1995)].